TY - JOUR

T1 - Character sums and the series L(1,χ) with applications to real quadratic fields

AU - Leu, Ming Guang

PY - 1999

Y1 - 1999

N2 - In this article, let k = 0 or 1 (mod 4) be a fundamental discriminant, and let x(n) be the real even primitive character modulo k. The series can be divided into groups of k consecutive terms. Let v be any nonnegative integer, j an integer, 0 ≤ j ≤ k - 1, and let Then. In section 2, Theorems 2.1 and 2.2 reveal a surprising relation between incomplete character sums and partial sums of Dirichlet series. For example, we will prove that T(v, j, χ) · M < 0 for integer if and |M | ≥ 3/2. In section 3, we will derive algorithm and formula for calculating the class number of a real quadratic field. In section 4, we will attempt to make a connection between two conjectures on real quadratic fields and the sign of T(0, 20, χ).

AB - In this article, let k = 0 or 1 (mod 4) be a fundamental discriminant, and let x(n) be the real even primitive character modulo k. The series can be divided into groups of k consecutive terms. Let v be any nonnegative integer, j an integer, 0 ≤ j ≤ k - 1, and let Then. In section 2, Theorems 2.1 and 2.2 reveal a surprising relation between incomplete character sums and partial sums of Dirichlet series. For example, we will prove that T(v, j, χ) · M < 0 for integer if and |M | ≥ 3/2. In section 3, we will derive algorithm and formula for calculating the class number of a real quadratic field. In section 4, we will attempt to make a connection between two conjectures on real quadratic fields and the sign of T(0, 20, χ).

KW - Character sum

KW - Dirichlet series

KW - class number formula

KW - real quadratic field

UR - http://www.scopus.com/inward/record.url?scp=0039445948&partnerID=8YFLogxK

U2 - 10.2969/jmsj/05110151

DO - 10.2969/jmsj/05110151

M3 - 期刊論文

AN - SCOPUS:0039445948

SN - 0025-5645

VL - 51

SP - 151

EP - 166

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

IS - 1

ER -